Curl and Convective Derivative

[Hercuflea]

Suppose u is a vector-valued function. Is it true that
(∇×u)⋅( (u⋅∇)u ) = (u⋅∇)(∇×u)⋅u
?

Please note the lack of a dot product on the first two terms of the RHS and the parenthesis around the second term of the LHS. I'm trying to understand whether these differential operators are associative.

 

[andrewkirk]

There will be two ways of doing this. One is using existing vector calculus identities, if there are ones that would be helpful. In the absence of that, one has to fall back on plan B: write it out in full in terms of coordinates, using
$$\nabla=\frac{\partial}{\partial x}+
\frac{\partial}{\partial y}+
\frac{\partial}{\partial z}$$

$$\mathbf{a\cdot b}=\sum_{k=1}^3 a_kb_k$$
and likewise the coordinate definition of curl.

My guess is that it's correct, but the only way to be sure is to wade through the algebra.